Singular Limits of Relative Entropy in Two Dimensional Massive Free Fermion Theory

We show that certain singular limits of relative entropy of vacuum states in two dimensional massive free fermion theory exist and compute these limits. As an application we show that the c function computed by Casini and Huerta based on heuristic arguments arise naturally as a consequence of our results.


Introduction
von Neumann entropy is the basic concept in quantum information and extends the classical Shannon's information entropy notion to the non commutative setting. The role of entropy in Quantum Field Theory is more recent and increasingly important, and appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc. (see for instance [2,3,12,13,28] and refs. therein).
The singular limits of certain relative entropies contain rich information about the underlying QFT. See Section 4 of [15] for an example where the global index of the underlying conformal net appears. Another such example is the paper [3] in which the celebrated c-theorem in 2 d QFT was derived from these singular limits (assuming they exist) by using monotonicity and covariance. The paper [3] contains physical arguments for c-theorem, and in the beginning of Sect. 2 we give a brief discussion. We refer interesting readers to the paper [3] for more details. It is clear that these arguments used certain unproved assumptions about singular limits of relative entropy. These singular limits are described precisely in Section 2, and is the main focus of this paper. To describe our main results, let us first recall some work in physics literature. In [5,7] Casini-Huerta computed c function in the case of 2d free massive fermions from singular limits of relative entropy and obtained the remarkable result that such function can be expressed in terms of functions which are related to solutions of Painleve equations of type V (cf. Eq. 31). However the starting point of [5,7] is the replica trick which uses formal manipulations of a density matrix that does not exist. There are identities in [4] where both sides are infinite, and in deriving an integral formula for the c-function the authors used integration by parts on these infinite functions, assuming certain vanishing properties on the boundary of the integral (cf. Remark 6.12). In fact it is not even known that the function that appears in formula (148) of [5] is integrable. This makes it very challenging to justify the formulas in [5]. See the second paragraph in the introduction of [14] for recent comments.
In this paper we take the point of view that relative entropy is fundamental in QFT, and by investigating the properties of their singular limits we should recover the above mentioned formulas in [5]. Our main results (cf. Th. 6.8, Th. 6.11 and Cor. 6.13 ) give an explicit formula for the singular limits in terms of τ function, and as a consequence (cf. Cor. 6.14) the formula of c-function in [5] follow. For reader's convenience, here is an explicit formula for singular limits of relative entropy as described in Cor. 6.13: the singular limits of relative entropy as defined in Eq. (6) where intervals are as in Fig. 6 are given by the following formula and τ 0 (t, β) is as defined after Eq. (32).
To prove these results, one of our key observations is that the deep results of [22,23] provide the right mathematical framework to investigate the properties of the singular limits, and indeed our theorems are proved in such framework, and our earlier results in [26] and [29] also play a crucial role. Even though the starting point of our computation is different from that of [5], some of the ideas are similar to that of [5], since the computations of Green's functions in [5] are special cases of the general theory of [22,23]. Here are more detailed account of our results: (1) in Sect. 3 we show that the singular limits exist based on a result in [25]; (2) we show that suitably modified two point "wave function" of [22,23] gives the resolvent of an operator, and this gives a formula to compute the singular limits; (3) by using the results of [22] and [23] we reduce the computation of the trace of kernel to a local computation around the ends of the interval, allowing us to express the singular limits in terms of τ function. Along the way we also find some seemingly new properties of τ function (cf. Remark 6.10).
The rest of this paper is organized as follows: In Sect. 2 we define the singular limits we want to compute. In Sect. 3 we show that the singular limits defined in Sect. 2 exist and analyze their properties. In Sect. 4 we collect the results from [22] and [23] that we will use in the paper, and also to set up notations. In Sect. 5 we prove a resolvent formula which will enable us to compute the singular limits. In Sect. 6 we use the results from the previous sections to give explicit formula for these singular limits. As a consequence in the last Sect. 6.3 we derive the formula of Casini-Huerta for c-function.
There are many interesting questions left open by this work. It will be interesting to extend our computation of singular limits to more general non-equal time slice configurations. We expect that the deep results of [23] will be useful. In the case of free bosons, since the entropy formula involves unbounded operators, our methods do not apply immediately. Finally, it seems to be a challenging question to have well controlled singular limits in more than two dimensional space time, since the singularities are no longer of the logarithm type, and the dependence of these singularities on the boundary is more complicated. We expect our results and ideas may be useful in addressing these questions.
We'd like to thank E. Witten for stimulating discussions.

The Singular Limits of Relative Entropy
In this section we describe the question that will be addressed in this paper. First we will make some general comments. See [11] for more details on algebraic formulation of QFT is Araki's relative entropy of the two states (cf. Section 2.1 of [15]). It is expected (cf. [19]) that F(O 1 , O 2 ) is finite and goes to infinity when O 1 and O 2 get closer to each other. We will focus in the two dimensional case in this paper, and the double cones lie on time equal to 0 slice. In that case we can represent double cone by its time equal to 0 slice. Remark 5.11 for precise statements in the massive free fermion case). The singular limits in this case are already interesting (cf. (2) of Th. 4.2 in [15]). As already observed in [3] (also cf. Section 4.2 of [15] Fig. 1, then when c 2 → c + heuristically the logarithm divergence will cancel out at the end c and we may have a well defined function, denoted by F(b 1 , b, c, c 1 ). Casini and Huerta argued in [3] that such function can be written as where G(t), t > 0 is "renormalized" entropy. Let t = e s , and define the c function to be c(s) = dG(e s ) ds . Casini and Huerta argued that c(s) ≥ 0 and c (s) ≤ 0 based on covariance and monotonicity of relative entropy. In Section 4.2 of [15] a large class of such functions in the case of conformal field theories are determined which have c function equal to a constant proportional to the central charge.
Our goal in this paper is to show that lim c 2 →c + F(b 1 , b, c; c 2 , c 1 ) exists and compute the limiting function F(b 1 , b, c, c 1 ) in the case of massive free fermions. We refer the reader to the introduction part of the paper [8] where general free fermion theory is described in Algebraic QFT framework. We will only be interested in the case of massive free fermions in two dimensional Minkowski spacetime. In the following we will introduce the formula for relative entropy F(b 1 , b, c; c 2 , c 1 ) as discussed on Page 1475 of [29]. First we will do some preparations. Specializing the formulas in the introduction part of the paper [8] to the case of massive free fermions in two dimensional Minkowski spacetime, our problem can be formulated as follows (also cf. [6]). Let where K 0 , K 1 are modified Bessel functions of second kind. Let where I 2 is two by two identity matrix.
Denote by C the operator on L 2 (R, C 2 ) given by Here we have chosen somewhat unconventional notation by integrating with respect to the first variable. The reason will be explained in Sect. 5.4. We write By the paragraph after Eq. (14), C is a projection. To compare with the notation in [8], this is the projection that is denoted by P in equation (10) of [8] that defines the Fock state which is the ground state for 2-dimensional Minkowski space-time. The reader can also find similar formula in equation (64) of [6].
If I is a closed interval, we will denote by P I the projection on L 2 (R, C 2 ) which is given by multiplication by the characteristic function of I . In Fig. 1 we have used 1, 2, 3 to denote the intervals Introducing the following: Definition 2.1.Ĉ I := P I C(1 − P I )C P I . When x is in the resolvent set of the operators T,Ĉ 12 ,Ĉ 23 ,Ĉ 2 ,Ĉ 123 , we define When it is necessary to indicate the dependence ofĈ(1 ∪ 2, 2 ∪ 3, x) on the intervals, we will writeĈ(b 1 , b, c; c 2 , c 1 , x). Let 0 ≤ E ≤ 1 be an operator. Let R E (β) = (E − 1/2 + β) −1 . Note that R E (β) is holomorphic in β ∈ C − [1/2, −1/2] in the operator norm topology. It is now useful to recall the following formula (cf. [5]): By Lemma 3.12 of [26] the above formula is more conveniently written as . This follows as a direct computation as in Lemma 3.12 of [26] that By the first paragraph on Page 1475 of [29], the relative entropy is given by the trace of an operator constructed from the covariance operator as in definition (2). Following the definitions in a straightforward way we have the following formula The main question in this paper to determine lim c 2 →c + F(1 ∪ 2, 2 ∪ 3) and study its properties.
In Sect. 6 we will frequently encounter functions H (a, b, w) or operators where a, b are end points of interval [a, b] and w denote the other variables H may depend on. We make the following: For the intervals as in Fig. 1, H (12 When it is necessary to specify the dependence of H (12, 23, w) on the end points of intervals, we will write H (12, 23, w)

Schatten-von Neumann ideals.
This paper relies on the results for general quasinormed ideals of compact operators. Here we limit our attention to the case of Schattenvon Neumann operator ideals S q , q > 0. Detailed information on these ideals can be found e.g. in [20] and [24]. We shall point out only some basic facts. For a compact operator A on a separable Hilbert space H , denote by s n (A), n = 1, 2, ... its n-th singular values, that is, the eigenvalues of the operator |A| := √ A * A. Note that if R 1 , R 2 are bounded operators, Then (cf. [24]) where ||A|| to denote the norm of an operator A. We denote the identity operator on H by 1. The Schatten-von Neumann ideal S q , q > 0 consists of all compact operators A, If q ≥ 1, then the above functional defines a norm; if 0 < q < 1, then it is a so-called quasi-norm. There is nevertheless a convenient analogue of the triangle inequality, which is called the q-triangle inequality: We also have the Holder inequality: See [17] and also [2]. In what follows we focus on the case q ∈ (0, 1]. We will use ||A|| to denote the norm of an operator, and ||A|| 1 the trace of |A|. By definition [25]. Let p(x, y, ξ) be a smooth function on R 3 . Define a class of operators on L 2 (R 1 ) with symbol p(x, y, ξ) as follows (10) where u ∈ L 2 (R 1 ). Let r > 0. Denote by L r loc (R 1 ) the set of measurable functions h such that |h| r is integrable on bounded measurable sets. 0 < δ ≤ ∞. Let C n be the unit interval centered at n. The lattice square norm is defined by
where C(q, m, 0 ) is a constant which only depends on q, m, 0 .
Proof. Ad (1): First note a 0 (ξ ) has no singularity for finite ξ . When ξ > m, we have the following convergent series expansion for a 0 (ξ ) It follows that when ξ is sufficiently large we have ∂ k a 0 ( Similarly the same bound holds when −ξ is sufficiently large. When q(k + 1) > 1, the series |n|>0 1 n q(k+1) < ∞, and (1) is proved. (2) is proved similarly. Corollary 3.3. Assume that h i = χ E i , i = 1, 2 are characteristic functions of measurable sets E i , and the distance between E 1 , E 2 is greater or equal to 0 > 0. Let a 0 (ξ ), a 1 (ξ ) be as in Lemma 3.2. Then for any q ∈ (0, 1], i = 1, 2 we have whereṽ l are defined as in Eq. (19).
where I 2 is two by two identity matrix. Note that the Fourier transform ofṽ 0 (x) is π 2 a 0 (ξ ), and the Fourier transform ofṽ 1 (x) is −i π 2 a 0 (ξ ) (cf. Section 6.5 of [9]) as in Lemma 3.2. In the momentum space or after Fourier transform (note that Fourier transformation maps convolution into products) the operator D m is simply multiplication by Using a 1 (ξ ) = ξ √ ξ 2 +m 2 , a 2 1 + m 2 a 2 0 = 1 we see that D m is self adjoint and D 2 m = 1/4. Denote by C the operator on L 2 (R, C 2 ) whose kernel is given by C(x, y) in Eq. (2). We write Using the fact that D m is self adjoint and D 2 m = 1/4 we see that C is a projection. Let h 1 , h 2 be characteristic functions of two measurable sets whose distance is greater or equal to 0 . By inequality (8) we have Since up to constants each C i j is Op(a 0 ) or Op(a 1 ), by Cor. 3.3 we have proved the following: (2) Proof.

Split of intervals.
The intervals are as in Fig. 1, where 1 : We assume that the length of intervals 1, 2 are bounded from below by 0 . Note that we allow b 1 (resp. c 1 ) to be −∞ (resp. ∞). Fix 1/2 < σ < 1. Given two operators T 1 , T 2 , we write T 1 ∼ T 2 if |T 1 − T 2 | S σ ≤ C( 0 ) for some constant C( 0 ) which only depends on 0 . Note that inequality in (8) implies that if T 1 ∼ T 2 , T 2 ∼ T 3 , then T 1 ∼ T 3 and similarly if T 1 ∼ T 2 , T 3 ∼ T 4 , then c 1 T 1 + c 2 T 2 ∼ c 1 T 1 + c 2 T 2 for any bounded operators c 1 , c 2 by Eq. (9). Also note that T 1 ∼ T 2 iff T * 1 ∼ T * 2 . For two subset E, F of the real line we use d(E, F) to denote the distance between them.
is a constant which only depends on 0 , and z * > 0 is defined as in Definition 3.5.
Proof. The theorem is proved by a series of reductions, and in each step we will throw away terms T which verifies ||T By Lemmas 3.6 and 3.7, after thrown away a term T which verifies ||T || 1 ≤ 3.9, Lemmas 3.6 and 3.7 we can replace g 0 (P 23 C(1−P 123 )C P 23 , z) by g 0 (P 6 C(P 0 + P 4 )C P 6 , z). Similarly by Prop. 3.9, Lemmas 3.6 and 3.7 we can replacê C 23 by P 5 C P 1 C P 5 + P 6 C(P 0 + P 4 )C P 6 . Hence we can replace Note that this term is independent of interval 3. Hence by repeating the above reductions for while taking 3 to be an empty set, and recall thatĈ(1∪2, 2∪3, z) as defined in Definition 2.1, we have proved the theorem. Fig. 2, dropping −∞ we will write the correspondingĈ (12, 23, x) Fig. 2, we will write the correspondinĝ Fig. 2. (1) follows by Lemma 3.10 of [15]. (2) follows from (1) and definitions.

Proposition 3.13. Assume that z is a complex number with z
Proof. All equations are proved in the same way. Let us prove the first equation. It is enough to examine the "thrown away terms" in the proof of Th. 3.10. By Lemma 3.6, all such "thrown away terms" are as in (1) of Lemma 3.7. A typical such term is of the following and either P i or P j depends on y = c 2 − c, and this projection, denoted by P(y) decreases or increases to a projection P strongly as y → 0 + . For definiteness assume that P j = P(y). We can replace P(y) by P(y)P(η) when y < η and P(y) is decreasing, or P(y) by P(y)P when P(y) is increasing. Apply Lemma 3.12 to g 1 (y)P i C P(η)P(y)g 3 (y) with A = P i C P(η), or to g 1 (y)P i C P P(y)g 3 (y) with A = P i C P, note that in both cases by Cor. 3.4 A is trace class, we conclude that when y → 0, the thrown away terms inĈ(b 1 , b, c; c 2 , c 1 , z) converges to the corresponding thrown away terms inĈ(b 1 , b, c, c 1 , z) in trace, and similarly for the rest of equations.
Recall that 1/2 < σ < 1. Since the function 2β the theorem now follows from Dominated Convergence Theorem and Prop. 3.13.
Proof. As in the proof of Prop. 3.13, it is enough to show that the trace of "thrown away term" in the proof of Th. 3.10 is holomorphic in where A is a fixed trace class operator, and f 1 (z), f 2 (z) are holomorphic in z with respect to the norm topology since z ∈ C−[−1/4, 0]. By Lemma 3.12 the proposition is proved.
By Eq. (6) we have the following formula We will compute this function in Sect. 6.

Some Results from [22,23]
In this section we will review some of the results of [22] and [23] that will be used in this paper. The reader is encouraged to consult [22] and [23] for more details. For a survey, see [21]. The functions introduced at the end of this section will play a crucial role in Sects. 5 and 6. Unfortunately there is no clear rigorous short cut to explain why these functions may be useful for our purpose. See [18] for a different perspective which may be helpful. Let with positive mass m > 0. We introduce a series of multi-valued special solutions of the above equations. For l ∈ C let I l (x) and K l (x) denote the modified Bessel functions of the first and second kind respectively. Set where z = 1 2 re iθ ,z = 1 2 re −iθ , r ≥ 0, θ ∈ R. These functions are multi-valued solutions (outside the origin) of Eq. (18), having the following local behavior as |z| → 0: where l! = (l + 1), and ... are higher order term. We also havẽ where γ 0 is Euler's constant. These functions satisfy the following recursion relations where ∂ := ∂ z ,∂ := ∂z. In this paper we will make use of two point "wave functions" as defined in section 3.2 of [22]. Let (a i ,ā i ), i = 1, 2 be two distinct points of R 2 . Denote byX the universal covering manifold of 2 with the covering projection π :X → X . We fix base pointsx 0 ∈X , x 0 ∈ X so that π(X 0 ) = x 0 , and denote by π 1 (X ; x 0 ) the fundamental group of X . An element γ ∈ π 1 (X ; x 0 ) is identified with the covering transformation it induces onX . For a function u onX we ,a 2 will denote the space consisting of complex valued real analytic function v onX with the following properties: where k is ±l ν + j, c s are constants independent of z, and the sum is absolutely convergent on any compact subset of |z − a ν | < η; We note that our W l 1 ,l 2 a 1 ,a 2 is W l 1 ,l 2 ,strict B,a 1 ,a 2 on page 589 of [22]. Denote by L the matrix l 1 0 0 l 2 and by slightly abusing notations we denote by 1 − L the matrix For each such fixed L We will make use of two canonical elements in W l 1 ,l 2 a 1 ,a 2 (cf. Page 592 of [22] where ... are higher order terms. Here we have suppressed the dependence of v μ (L) on z,z, a μ ,ā μ , μ = 1, 2, and the dependence of α μν (L), β μν (L) on a μ ,ā μ , μ = 1, 2 in the notation when no confusion arises.
Only when it is necessary to indicate the dependence of α μν (L), β μν (L) on a ν ,ā ν we will write them as α μν (a 1 , a 2 , L), β μν (a 1 , a 2 , L). We shall make use of the next two lemmas in Sect. 5.
Proof. Let us first consider the case when 1/2 > l μ > 0, μ = 1, 2. By Prop. 3.1.3 of [22] we have the following expansion of v around a μ : In both cases we see that v ∈ W l 1 ,l 2 .
Proof. By definition around a 1 v has expansion where ... are higher order terms. Note that when 0 where in the first inequality δ = −l μ modZ, and in the second inequality δ = l μ modZ, l μ , μ = 1, 2 are not integers, and C is a constant independent of r .
Proof. The proof is implicitly contained in the proof of Prop. 3.1.3 of [22]. We will prove the first inequality since the second one is proved in a similar way. As in the proof of Prop. 3.1.3 of [22], let z − a μ = 1 2 re iθ , we have We have The second inequality is proved similarly.
Recall that L is the matrix We use the following notation The next proposition follows from Prop. 3.2.5 in [22]: The entries of α(L), β(L) satisfy a remarkable set of equations. These equations are derived by considering the deformation of the equations satisfied by v μ (L) in [22] and more generally by using product formula in [23]. Let us now describe these equations as on Page 935 of [23] (also cf. Page 623 of [22]). Let us first define an invertible matrix G whose inverse is given by β(L) sin(π L) = −G −1 . Let Then G takes the following form where Then β(L) sin(π L) = −G −1 , and G −1 takes the following form k := k l (t), ψ := ψ l (t) only depends on l, t, where k l (t), ψ l (t) are as in Eq.(29), k, ψ satisfy the following: The above equation for ψ can be converted to the following Painleve equation of the fifth kind by the substitution s = t 2 , σ = tanh 2 (ψ):

τ Function and related functions.
Let us also recall some basic facts about τ function. We will only consider the case when l 1 = −l 2 , l = 2l 1 . This is a function which is analytic in l when |l| is small (cf. Lemma 4.11), and real analytic in a i ,ā i , i = 1, 2. The defining equation is (cf. Remark on Page 628 of [22] or Page 936 of [23]): Equation (32) will play an important role in Section 6.2. Note τ 0 := ln τ only depends on t as in Eq. (27) and l, and we will write such a function as τ 0 (t, l), and suppress its dependence on t, l in the following for the ease of notations. Let τ 1 = −τ 0 . Then by equation (4) in [27] τ 1 satisfies the following equation: ψ satisfies the Eq. (30) with boundary condition as t → ∞. We note that when l 1 ∈ R, 4i sin(πl 1 )K 2l 1 (t) ∈ iR and when l 1 ∈ iR, 4i sin(πl 1 )K 2l 1 (t) ∈ R. We have (cf. Appendix of [7]) ψ ∈ iR when l 1 ∈ R, and ψ ∈ R when l 1 ∈ iR.
Remark 4.7. In Sect. 6 we will sometimes consider real analytic function f (z, ...) restricted to z ∈ R, where ... stands for possible other variables, and consider the partial derivative of f with respect to the real variable z. To avoid possible confusions with ∂ z which is defined to be the partial derivative of f with respect to the complex variable z, we will use d f dz to denote the partial derivative of f with respect to the real variable z. We also have the following facts from [22]: where c, are as in Eq. (27); (2) When l 1 = −l 2 ∈ R, we haveβ 21 (L)β 21 (1 − L) = sinh 2 ψ; (4) When a 1 , a 2 are real and let t = 2m(a 2 − a 1 ) > 0, then we have Here we have used dα 11 da i to denote the partial derivative of α 11 with respect to the real variable a i as in Remark 4.7.
When |l| = 2|l 1 | is sufficiently small, we have the following convergent power series expansion for τ 0 = ln τ (cf. Page 933 of [23]): and u ±l k stands for u −l k for even k and u l k for odd k, λ = i sin(πl 1 ).
, and K is the transpose of K . When λ 1 is sufficiently small we have So we see that τ 0 = −τ 1 = −τ 2 . For an operator T we use ||T || and ||T || 2 to denote its norm and Hilbert-Schmidt norm respectively. Lemma 4.9. Assume that t ≥ η > 0, |l| < 1. Then Proof. Fix u > 0, let us evaluate the maximum of ∞ 0 |K (u, v)|dv. Denote by r the real part of l. Since |r | < 1 We have To find the maximum of f r (u), u > 0, it is sufficient to assume that 1 > r ≥ 0. Since . One easily checks that the maximum of f 1 (u) is a function of η, and goes to 0 as η → ∞.
Similar estimate holds when we integrate with respect to u and fix v in the previous paragraph. By Th. 6.18 of [10] we have proved ||K || ≤ C 0 (η) where C 0 (η) → 0 as η → ∞. We note that ||K || 2 | is a decreasing function of t and is dominated by an integrable function independent of t for any t ≥ η. Since |K (u, v)K (v, u)| goes to 0 as t → ∞, Lebesgue's dominated convergence theorem implies that ∞ 0 ∞ 0 |K (u, v)K (v, u)|dvdu goes to 0 as t → ∞. The proof for K is similar.
Given two series n≥0 f n , n≥0 g n we say n≥0 f n is dominated by n≥0 g n if | f n | ≤ |g n | for all n ≥ 0.
The following Lemma follows from Cauchy's formula and the proof can be found in If f (z) = n≥0 f n (z) and assume that n≥0 f n (z) is dominated by a series n≥0 C n with C n independent of z and n≥0 |C n | < ∞. Then f (z) = n f n (z) and the series converges absolutely and uniformly on any compact subset of U .

Proof. Ad (1): Note that
then the series is dominated by a convergent geometric series. It follows immediately that τ 0 → 0 as t → ∞ or as l → 0, and τ 0 is a holomorphic function of l 1 for |l| < C 0 (η) .
By (1) where by Lemma 4.10 the series on the righthand side converges absolutely and uniformly on any compact subset of |l 1 | < C 0 (η) . To prove the rest of cases in (2) and (3), it is sufficient to check each term ∂ l 1 ( (2i sin(πl 1 )) 2k k e (2k) l (t)) has the properties described in (2) and (3). For an example since ∂ l 1 (2i sin(πl 1 )) 2k → 0 as l 1 → 0, and (2i sin(πl 1 )) 2k → 0 as l 1 → 0, we have ∂ l 1 τ 0 → 0 as l → 0. The rest of the cases are proved in a similar way. Here c k is independent of r and is holomorphic in l 1 , l 2 . As on page 587 of [22] we have The term K 0 (mr) can be estimate directly as K 0 (mr) = O( 1 √ mr e −mr ), and (2) is proved.

The Resolvent
Let I := [a 1 , a 2 ] ∈ R be a closed interval. Throughout this section we shall assume that the following conditions are satisfied: where C 0 (η), C 2 (η) are constants in Lemmas 4.11 and 4.12 respectively.

Condition H .
We first recall some results from Chapter 2 of [17].
The proof of the following formula, also known as Sokhotski-Plemelj formula, can be found on Page 49 of [17]: = (a 1 , a 2 ). Then if x ∈ (a 1 , a 2 )

Lemma 5.2. Assume that f (x) verifies condition H on I
We will also need the property of singular integrals with Cauchy kernels at the ends of the interval I ( cf. Page 74 of [17] for more precise estimates): where A does not depend on x 1 , x 2 but may depend on [a 1 , a 2 ], 0 < μ < 1.
(2) Assume that near the ends c = a 1 or c = a 2 the function φ(x) is of the form where φ * (x) satisfies condition H near and at c.
x−z . First assume that z is near c but not on I . Then (3) If 2πi ln 1 z−c + 0 (z) where the upper sign is taken for c = a 1 , the lower for c = a 2 . ln 1 z−c is to be understood to be any branch, single valued in the plane cut along I ; 0 (z) is a bounded function; (4) 2i sin γ π + 0 (z) where the signs are as in (3), (z − c) γ is any branch, single valued near c in the plane cut along I and taking the value (x − c) γ on the upper side of I , and | 0 (z)| < C |z−c| α 0 where C, α 0 are real constants with α 0 < α. For the point z = x 0 lying on I , the following results hold: (5) 2πi ln 1 z−c + 0 (x 0 ) * where 0 (x 0 ) * satisfies the condition H near and at c, and the signs are again as in (3) where the signs are as in (3), 0 (x 0 ) = * * (x 0 ) |x 0 −c| α 0 where * * (x 0 ) satisfies condition H near and at c and α 0 < α.
We will also make use of the following  1 4 . By a result on Page 17 of [17] for fixed x ∈ I , |ψ( that ψ(x 1 , x), |x − x 1 | μ and |x − x 2 | μ are bounded functions on I and we will denote by M a constant which is greater than the upper bounds of these three functions.
We have where M is constant independent of x. By using Holder inequality we have Note that by Holder inequality again we have I |x 1 −x| −2μ |x 2 −x| −2μ dx ≤ ( I |x 1 − x| −4μ dx) 1/2 ( I |x 2 − x| −4μ dx) 1/2 < M where M is a constant independent of x 1 , x 2 since μ < 1/4, and the Lemma is proved.

Solving the linear equation. Recall from definition (1)
where I 2 is two by two identity matrix. We denote by D 0 (y, x) the massless limit of D m (y, x) and it is given by (48) A key idea to find resolvent for the operator defined as in definition (1) is to extend x from I to the complex plane. This is well defined becauseṽ i (y − z), i = 0, 1 are real analytic function of z,z when z = y. More precisely we define (50) Then ∀x ∈ (a 1 , a 2 ) Proof. Fix x ∈ (a 1 , a 2 ). We have D m (y, z) = D 0 (y, z) z) is a smooth function, and E 3 (y, z) is ln |z − y| multiplied by a smooth function F 3 (y, z). We have Notice when y = x, the integrand converges pointwise to 0 as → 0 + . For fixed x, the integral over y can be divided into two parts: the first part is over those y ∈ I with |y − x| ≥ 1/2, and the second part is over y ∈ I with |y − x| ≤ 1/2. On the first part of the integral the integrand can be dominated by a constant, and on the second part of the integral, when is small than 1/2, |x − y| 2 + 2 < 1, and hence the integrand is dominated by a constant multiplied by | ln |y −x|g(y)|, which is an integrable function on I . By Dominated Convergence Theorem we have lim →0 It is enough to prove the first equation in the Lemma for D 0 (y, x), and this follows immediately from Lemma 5.2. The second equation also follows in a similar way. Notice that the last equation follows from the first two by solving

Resolvent in
the case of m = 0. This case is a well known case of Riemann-Hilbert problem and more general case is discussed for an example on Page 130 of [16]. We will give a sketch of proof in our case.
It is sufficient to check the equation on smooth functions. Let g(x) be a smooth function on I . We will solve the following linear equation: Since g is smooth, f is smooth on (a 1 , a 2 ). Recall that h(z) = (z−a 1 ) −l (a 2 −z) l , and we have chosen the branch cut to be [a 1 , a 2 ] and define h(z) so that h( ∈ (a 1 , a 2 ). ∈ (a 1 , a 2 ) and G l (y, x − i0) = e −2πil G l (y, x), ∀x = y ∈ (a 1 , a 2 ).
We note that G l (y, z) satisfies many of the properties of v 0 in Eq. (40). For an example, ∂ a 1 G l (y, z) = l(z − a 1 ) −l−1 (a 2 − z) l (y − a 1 ) l−1 (a 2 − y) −l is non-singular at z = y, and factorized as products of a function of y and a function of z, similar to Eq. (43). This strongly suggests that the resolvent for m > 0 case can be obtained from suitable functions of v 0 , and we will prove in the next section that is indeed the case.

The general case.
Recall the function v 0 (z * , z, L) defined at the end of Sect. 4. Consider the following function Note that by definition R m (z * , z) also depends on L , a 1 , a 2 and we have suppressed its dependence on L , a 1 , a 2 in the following for the ease of notations. In fact, from now until the end of this paper, we make the following declaration: Starting from Sect. 5.4 until the end of this paper, whenever a function is defined, and it is clear from the definition that such a function depends on L (or equivalently l), a 1 , a 2 , unless otherwise stated, we will suppress its dependence on any of L (or equivalently l), a 1 , a 2 for the ease of notations following the convention in [22]. a 1 a 2

Fig. 4. A second cut
We come up with R m (z * , z) by demanding that R m (z * , z) share the following properties with D m (z * , z) as defined in Eq. (49): the second row is obtained from the first row by applying im −1 ∂z. Moreover the entries of R m (z * , z) as a function of z satisfy (1), (2) and (4) of definition 4.1.
We will now give a formula for the resolvent. We choose the cut of the complex plane to be I = [a 1 , a 2 ]. We choose a branch of R m (z * , z) on C − [a 1 , a 2 ], denoted by R m (z * , z) as follows. For y = x ∈ I , we define R m (y, x) := R m (y, x + i0) = lim →0 + R m (y, x +i ), i.e., the value of R m on the upper cut of I . Define R m (y, x −i0) = lim →0 − R m (y, x − i ), i.e., the value of R m on the lower cut of I . Note that from Eq. (53) and definition of v 0 (z * , z, L + 1) before Eq. (39) we have R m (y, x − i0) = e −2πil R m (y, x + i0). By Eq. (39) we can choose R m (y, x) such that when x is close to y, and y = x, R m (y, x) = D m (y, x)+ non-singular terms, where both x, y are in (a 1 , a 2 ). Since l 1 = −l 2 , R m (z * , z) is a well defined single valued function for z = z * , and both z, Let R m be the integral operator on L 2 (I, C 2 ) given by the kernel R m (y, x), i.e, Note that the notation R m follows the declaration after Eq. (53). Here we have chosen somewhat unconventional notation by integrating with respect to the first variable in R m : this is in tune with the notation of v 0 (z * , z, L), where z * corresponds to integration variable. The most singular part of R m (y, x) when x − y → 0 is (up to multiplication by constants) 1 x−y and the integral is understood as Cauchy's Principle Value. When x = y and both in (a 1 , a 2 ), R m (y, x) is a smooth function of y.
For fixed x ∈ (a 1 , a 2 ), by Eq. (40) near a i R m (y, x) has the worst singularity of the form (x − a i ) −l when l > 0, i = 1, 2.
Let f (y) be a smooth function on I . LetĜ(z) := 1 β+1/2 I R m (y, z) f (y)dy. Note that the notationĜ(z) follows the declaration after Eq. (53). G(z) verifies (1), (2) and (4) of definition 4.1 on the plane minus the cut I , since R m (y, z) as a function of z does when z = y. The following gives the properties ofĜ(z) near and on the cut I : Lemma 5.7. ∈ (a 1 , a 2 ).
Proof. Let us choose a second cut on the complex plane as given by the three line segments below I joining the ends of I in Fig. 4. Let S 1 (z * , z) be the function from R m (z, z * ) in Eq. (53) defined on this second cut, such that S 1 (y, z) = R m (y, z) when Imz > 0, y ∈ I . When we move z from above [a 1 , a 2 ] to the region between the two cuts (the interior of the square in Fig. 4), we have crossed the cut for R m (y, z), and by definition we have that when z is in the region between the two cuts, 1 (y, z).

It follows that
Note that when y is close to x, by Eq. (39) that gives the asymptotics when y is close to x, and definition of D m , S 1 (y, The first equation now follows as in the proof of Lemma 5.5. For the second equation, choose η > 0 to be small and let I η := (x − η, x + η). When → 0 + , we have where we have used e 2πil = β−1/2 β+1/2 . On I η , up to terms which go to 0 as η → 0, we have lim where in the last = we have used Lemma 5.2, and o(1) denotes a term that goes to 0 as η → 0. The proof now is complete.
where I 2 is the identity two by two matrix.
Proof. We will check the resolvent formula on a dense subspace of L 2 (I ).
Let f (x) be a smooth function on I with support of f contained in (a 1 , a 2 ). We will solve the following linear equation: First since β ∈ iR, β = 0, β is in the resolvent of the self adjoint operator D m , and the above linear equation has a unique solution g ∈ L 2 (I ). Let us first show that g satisfies condition H on (a 1 , a 2 ). We re-write the above equation as Since the entries of D m (y, x)−D 0 (y, x), up to addition of a smooth function of (x, y), equal to ln |x − y| multiplied by a smooth function of (x, y), by Lemma 5.4 g 1 (x) := I (D m (y, x)− D 0 (y, x))g(y)dy satisfies condition H on I . Apply the resolvent formula in Th. 5.6 we have g 1 (y))dy.
Since f (x) − g 1 (x) satisfies condition H on I , by the result on Page 49 of [17] we see that g(x) satisfies condition H on (a 1 , a 2 ). By applying item (5) and (6) (1) and (4) in Definition 4.1 on the plane with cut I . Apply Lemma 5.5 to G 2 and Lemma 5.7 to G 1 for x ∈ (a 1 , a 2 ) we have Now we examine the growth properties of G 1 (z), G 2 (z) near the ends a i . By the second paragraph after Eq. (53) we can write the components of G 1 (z), β+1/2 , the support of f is contained in (a 1 , a 2 ), when z is close to a i , we only need to consider the property of R m (y, z) when |z − y| ≥ η 0 > 0 where η 0 is a fixed small constant. By Eq. (40), it follows that As for G 2 (z), note that D m (y, z) − D 0 (y, z), up to addition of smooth functions, is ln |z − y| multiplied by a smooth function. By Lemma 5.4 the integrals involving ln |z − y| are bounded around a i . So up to bounded functions around a i we can replace D m (y, z) by D 0 (y, z) in the definition of G 2 (z).
Apply (3) and (4) . Apply Lemma 5.5 to G 2 we have . Using G 1 = G 2 we have proved the theorem.

Some properties of the resolvent.
Recall from the definition of R m (z * , z, L) below Eq. (53) where we have put in extra L dependence. By definition R m (z * , z, L) also depends on a 1 , a 2 and we have followed the declaration after Eq. (53). We will use R m (L) to denote the corresponding integral operator. Recall from condition (45 Proof. By Th. 5.8 Since β ∈ iR and D m is self adjoint, we have where we have used e 2πil = β+1/2 β−1/2 . It follows that R m (L) H = R m (−L), where R m (L) H is the adjoint of integral operator R m (L), and the Lemma is proved.
It is convenient to introduce for fixed L Note that by definition R(z * , z) also depends on a 1 , a 2 , L and we have followed the declaration after Eq. (53). By equation (3.2.23) of [22] we have the complex conjugate of i π ∂ z v 0 (z, z * , L + 1) is i π ∂ z v 0 (z, z * , L). Note that by Eq. (39) singularities of v 0 (z * , z, L) and v 0 (z * , z, −L) canceled out at z = z * ∈ (a 1 , a 2 ). It follows that R m (z, z) is smooth for z ∈ (a 1 , a 2 ), and where tr 2 here means the sum of the diagonal entries of the two by two matrix. Using Eqs. (43) and (24) we have Similarly When a μ = x μ +iy μ with x μ , y μ real, we will be interested in computing z). Note that when a μ is real this is the same as m −1 d R(z,z) da μ . Following the convention in Remark 4.7 from Eqs. (57), (58) we have When a μ is real, and z ∈ (a 1 , a 2 ), from Lemma 5.9 R(z, z) ∈ iR. It follows that m −1 d R(z,z) da μ ∈ iR and we conclude that v μ (z, 1 − L)v μ (z, 1 + L) +v μ (z * , L)v μ (z, −L) is real. So when a μ , μ = 1, 2 are real we have the following equation By definition R(z, z) also depends on a 1 , a 2 , L and we have followed the declaration after Eq. (53). We note that R(x, x) is smooth in x, a 1 , a 2 when x ∈ (a 1 , a 2 ). The following Proposition gives information about the property of R(x, x) near the ends a 1 , a 2 .
Let us examine the expansion of πi sin(πl 1 ) v 2 (1 − L)v 2 (1 + L) around a 1 : by Eq. (23) we find that the first few leading order terms are, up to multiplication by constants which only depend on l 1 , α 21  The same argument works for the expansion around a 2 , with l 1 replaced by l 2 = −l 1 . (1) is proved. To prove (3), by Eq. (60) we have We should examine the expansion of v μ (z, 1 − L)v μ (z, 1 + L) + v μ (z, L)v μ (z, −L) and its complex conjugate around a 1 , a 2 . The proof is now the same as in the proof of (1).
Remark 5.11. From Prop. 5.10 if one integrate R(x, x) over [a 1 , a 2 ], we will get log divergence from the two ends a 1 , a 2 respectively. Moreover these singularities only depend on the ends a 1 , a 2 through l 1 , l 2 .

The constant term.
We remind the reader that we will follow the declaration after Eq. (53) for the rest of this paper.
Proposition 5.12. The constant term C 2 (μ) in (2) of Prop. 5.10 is given by Proof. It is sufficient to check m dC 2 (μ) da j , j = 1, 2 where we have followed the notation in Remark 4.7. We will check the case when μ = 1. μ = 2 is similar.
From Prop. 5.10 near a 1 we have the following expansion of R(z, z): By Eq. (44) m −1 ∂ z R(z, z) = −m −1 (∂ a 1 + ∂ a 2 )R(z, z), expanding both sides around a 1 , we see that the constant term on the LHS is C 3 , where the constant term on the where C 1 is the constant term of m −1 ∂ a 2 R(z, z) in its expansion around a 1 . It follows that m −1 ∂ a 1 C 2 (1) = −C 1 . On the other hand m −1 ∂ a 2 C 2 (1) is the constant term of m −1 ∂ a 2 R(z, z) in its expansion around a 1 , it follows that m −1 ∂ a 1 C 2 (1) = −C 1 = −m −1 ∂ a 2 C 2 (1). From the Eq. (57) Using the Eq. (23) for the expansion near a 1 , we find that the constant term C 1 of m −1 ∂ a 2 R(z, z) near a 1 is given by It follows that we find the constant term around a 1 is given by Usingβ(−L)β(L) = sinh 2 (ψ) in (2) of Prop. 4.8, we find We conclude from (4) of Prop. 4.8 that and the Prop. is proved. a 2 a 1 Fig. 5. Contour J , 1 : the two circles are oriented clockwise with radius and centers a 1 , a 2 respectively, and the two line segments are above and below the cut [a 1 , a 2 ] and 1 is the distance between them

The Computation of Singular Limits
Throughout this section except the last Sect. 6.3, we shall assume that the conditions in (45) are satisfied.
Let us first give a heuristic reason for our approach. We'd like to compute the singular limit F(12, 23) from (16) where 1, 2, 3 are intervals as in Fig. 6. Ignoring the singularities for the moment, we need to compute a 2 a 1 R(x, x)dx, where a 1 , a 2 are the end points of intervals in Fig. 6. Formally differentiating with respect to a 1 for an example, we get a 1 ). By Eq. (60) we need to compute )dx, its complex conjugate, and R(a 1 , a 1 ). It turns out with suitable linear combinations as in Sect. 6.2 the singularities will cancel out in the integrals, and the contribution from R(a 1 , a 1 ) is the constant term determined in Prop. 5.12. By Eq. (60) we should turn to the computation of

A reduction of a line integral to a local computation.
We will use I + , I − to denote the upper side of the cut [a 1 + , a 2 − ] and the lower side of the cut [a 1 + , a 2 − ] respectively with its usual orientation, namely from the left to the right. −J will denote the interval J with opposite orientation. The value of the functions on the left hand side of the integral in the next Lemma are by definition the value of such functions from the upper side of the cut [a 1 + , a 2 − ]. Lemma 6.1.
As in the proof of Prop. 5.10, the leading term in the expansion of v 1 (1 − L)v 1 (1 + L) around a 1 is given by −l 1 sin πl 1 π (mz − ma 1 ) −2 . This term gives the first term on the right hand side of the equation in (1). The next term is (mz − ma 1 ) 2l 1 or (mz − ma 1 ) −2l 1 . However when integrating such a term multiplied by iθ over |z − a 1 | = , up to constant it is bounded by 1 Proof. When μ = 1 this follows from (2) of Lemma 6.3 and Eq. (60), and the fact that both sides of (2) of Lemma 6.3 are real. μ = 2 is proved in the same way as the proof of (3) of Lemma 6.3.
For an interval I (our interval is connected), we will simplify our notation further by writing R(I, x) the (1, 1) entry of the resolvent R m (x, x, L) − R m (x, x, −L) (cf. Eq. 54) on the interval I when x ∈ I , and 0 when x is not in I . Recall from definition 2.2  2 ∪ 3, x), where intervals 1, 2, 3 are as in Fig. 6. Define   R 1 (b 1 , b, c, c 1 where we have followed the convention as in definition 3.11. As usual by definition R 1 (b 1 , b, c, c 1 ) also depends on l and we have followed the declaration after Eq. (53). Note the dependence of R 1 (b 1 , b, c, c 1 ) on its variables is very different from that of R m and we hope that there may be no confusions here. By Prop 5.10 R (12, 23, x) is continuous in the interior of intervals 1, 2, 3 and bounded near the ends of intervals 1, 2, 3.
From Eq. (16) Proof. We will prove the first equation. The second equation is proved in a similar way.
Let us compute . By definition Note that by (2) b 1 , c 1 , x) at b 1 + b 1 canceled out.